On the geometry of loop quantum gravity on a graph

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these and Regge geometries.Comment: 6 pages, 1 figure. v2: some typos corrected, references update

Geometrical constructions of equilibrium states

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such systems, we give in particular a constructive proof of the existence of the SRB measure and of the entropy maximizing measure. It is also established very fine statistical properties (Bernoulliness), and it is given a characterization of equilibrium states in terms of their conditional measures in the stable/unstable lamination, similar to the SRB case. The construction is applied to obtain the uniqueness of quasi-invariant measures associated to H\"older Jacobian for the horocyclic flow.Comment: Announcement of the results in arXiv:2103.07323, arXiv:2103.07333. 10 page

Introducing robocompass : a nifty tool for geometrical construction

How Geometrical Constructions are taught in Schools Euclid’s Elements – one of the most influential mathematical textbooks ever to have been written – is primarily a compendium of geometrical constructions created using straightedge and compass. But are we sure that these two geometrical tools which lie at the heart of such foundational ideas are being used effectively in the classrooms? The current practice is actually to use large wooden geometrical instruments in the classrooms, as the size of the real physical compass (in the ‘geometry box’) is not large enough to use conveniently as a demonstration tool

Geometrical Constructions of Flock Generalized Quadrangles

AbstractWith any flock F of the quadratic cone K of PG(3, q) there corresponds a generalized quadrangle S(F) of order (q2, q). For q odd Knarr gave a pure geometrical construction of S(F) starting from F. Recently, Thas found a geometrical construction of S(F) which works for any q. Here we show how, for q odd, one can derive Knarr's construction from Thas' one. To that end we describe an interesting representation of the point-plane flags of PG(3, q), which can be generalized to any dimension and which can be useful for other purposes. Applying this representation onto Thas' model of S(F), another interesting model of S(F) on a hyperbolic cone in PG(6, q) is obtained. In a final section we show how subquadrangles and ovoids of S(F) can be obtained via the description in PG(6, q)